The Dreams That Stuff is Made of
Page 55
If f (ξ, η) is any function of the canonical variables ξXk, ηk, the matrix representing f at any time t in the matrix scheme in which the ξ k at time t are diagonal matrices may be written down without any trouble, since the matrices representing the ξk and ηk themselves at time t are known, namely,
(2)
Thus if the Hamiltonian H is given as a function of the ξk and ηk, we can at once write down the matrix H(ξ’ ξ”). We can then obtain the transformation function, (ξ’/α’) say, which transforms to a matrix scheme (α) in which the Hamiltonian is a diagonal matrix, as (ξ’/α’) must satisfy the integral equation
(3)
of which the characteristic values W(αʹ) are the energy levels. This equation is just Schrödinger’s wave equation for the eigenfunctions (ξʹ/αʹ), which becomes an ordinary differential equation when H is a simple algebraic function of the ξk and ηk on account of the special equations (2) for the matrices representing ξk and ηk. Equation (3) may be written in the more general form
(3ʹ)
in which it can be applied to systems for which the Hamiltonian involves the time explicitly.
One may have a dynamical system specified by a Hamiltonian H which cannot be expressed as an algebraic function of any set of canonical variables, but which can all the same be represented by a matrix H(ξʹξ”). Such a problem can still be solved by the present method, since one can still use equation (3) to obtain the energy levels and eigenfunctions. We shall find that the Hamiltonian which describes the interaction of a light-quantum and an atomic system is of this more general type, so that the interaction can be treated mathematically, although one cannot talk about an interaction potential energy in the usual sense.
It should be observed that there is a difference between a light-wave and the de Broglie or Schrodinger wave associated with the light-quanta. Firstly, the light-wave is always real, while the de Broglie wave associated with a light-quantum moving in a definite direction must be taken to involve an imaginary exponential. A more important difference is that their intensities are to be interpreted in different ways. The number of light-quanta per unit volume associated with a monochromatic light-wave equals the energy per unit volume of the wave divided by the energy (2π h)ν of a single light-quantum. On the other hand a monochromatic de Broglie wave of amplitude a (multiplied into the imaginary exponential factor) must be interpreted as representing a2 light-quanta per unit volume for all frequencies. This is a special case of the general rule for interpreting the matrix analysis,jx according to which, if (ξʹ/αʹ)or ψaʹ (ξʹk) is the eigenfunction in the variables ξk of the state αʹ of an atomic system (or simple particle), |ψaʹ (ξʹk)|2 is the probability of each ξk having the value ξʹk, [or |ψaʹ(ξʹk)|2 dξʹ1dξʹ2... is the probability of each ξk lying between the values ξ’k and ξ’k + dξ’k, when the ξk have continuous ranges of characteristic values] on the assumption that all phases of the system are equally probable. The wave whose intensity is to be interpreted in the first of these two ways appears in the theory only when one is dealing with an assembly of the associated particles satisfying the Einstein-Bose statistics. There is thus no such wave associated with electrons.
§ 2. THE PERTURBATION OF AN ASSEMBLY OF INDEPENDENT SYSTEMS.
We shall now consider the transitions produced in an atomic system by an arbitrary perturbation. The method we shall adopt will be that previously given by the author,jy which leads in a simple way to equations which determine the probability of the system being in any stationary state of the unperturbed system at any time.jz This, of course, gives immediately the probable number of systems in that state at that time for an assembly of the systems that are independent of one another and are all perturbed in the same way. The object of the present section is to show that the equations for the rates of change of these probable numbers can be put in the Hamiltonian form in a simple manner, which will enable further developments in the theory to be made.
Let H0 be the Hamiltonian for the unperturbed system and V the perturbing energy, which can be an arbitrary function of the dynamical variables and may or may not involve the time explicitly, so that the Hamiltonian for the perturbed system is H = H0 + V. The eigenfunctions for the perturbed system must satisfy the wave equationih ∂ψ/∂t = (H0 + V)ψ,
where (H0 + V) is an operator. If ψ = Σrarψr is the solution of this equation that satisfies the proper initial conditions, where the ψr’s are the eigenfunctions for the unperturbed system, each associated with one stationary state labelled by the suffix r, and the ar’s are functions of the time only, then |ar|2 is the probability of the system being in the state r at any time. The ar’s must be normalised initially, and will then always remain normalised. The theory will apply directly to an assembly of N similar independent systems if we multiply each of these ar’s by N so as to make Σr |ar|2 = N. We shall now have that |ar|2 is the probable number of systems in the state r.
The equation that determines the rate of change of the ar’s iska
(4)
where the Vrs’s are the elements of the matrix representing V. The conjugate imaginary equation is
(4ʹ)
If we regard ar and ih arka as canonical conjugates, equations (4) and (4ʹ) take the Hamiltonian form with the Hamiltonian function F1 = Σrsarka Vrs as, namely,
We can transform to the canonical variables Nr , φr by the contact transformation
This transformation makes the new variables Nr and φr real, Nr being equal to arar* = |ar|2, the probable number of systems in the state r, and φr/h being the phase of the eigenfunction that represents them. The Hamiltonian F1 now becomes
and the equations that determine the rate at which transitions occur have the canonical form
A more convenient way of putting the transition equations in the Hamiltonian form may be obtained with the help of the quantities
Wr being the energy of the state r. We have |br |2 equal to |ar |2, the probable number of systems in the state r. For br we find
with the help of (4). If we put Vrs = vrs ei (Wr −Ws )t/h ; so that vrs , is a constant when V does not involve the time explicitly, this reduces to
(5)
where Hrs = Wrδrs + vrs , which is a matrix element of the total Hamiltonian H = H0 + V with the time factor ei(Wr−Ws)t/h removed, so that Hrs is a constant when H does not involve the time explicitly. Equation (5) is of the same form as equation (4), and may be put in the Hamiltonian form in the same way.
It should be noticed that equation (5) is obtained directly if one writes down the Schrödinger equation in a set of variables that specify the stationary states of the unperturbed system. If these variables are ξh, and if H(ξ’ξ") denotes a matrix element of the total Hamiltonian H in the (ξ) scheme, this Schrödinger equation would be
(6)
like equation (3’). This differs from the previous equation (5) only in the notation, a single suffix r being there used to denote a stationary state instead of a set of numerical values ξ’k for the variables ξk, and br being used instead of ψ(ξ’). Equation (6), and therefore also equation (5), can still be used when the Hamiltonian is of the more general type which cannot be expressed as an algebraic function of a set of canonial variables, but can still be represented by a matrix H (ξ’ξ") or Hrs.
We now take br and ih br* to be canonioally conjugate variables instead of ar and ih ar*. The equation (5) and its conjugate imaginary equation will now take the Hamiltonian form with the Hamiltonian function
(7)
Proceeding as before, we make the contact transformation
(8)
to the new canonical variables Nr, θr, where Nr is, as before, the probable number of systems in the state r, and θr is a new phase. The Hamiltonian F will now become
and the equations for the rates of change of Nr and θr will take the canonical form
The Hamiltonian may be written
(9)
The first term ΣrWrNr is the total proper energy of the assembly, and t
he second may be regarded as the additional energy due to the perturbation. If the perturbation is zero, the phases θr would increase linearly with the time, while the previous phases φr would in this case be constants.
§ 3. THE PERTURBATION OF AN ASSEMBLY SATISFYING THE EINSTEIN-BOSE STATISTICS.
According to the preceding section we can describe the effect of a perturbation on an assembly of independent systems by means of canonical variables and Hamiltonian equations of motion. The development of the theory which naturally suggests itself is to make these canonical variables q-numbers satisfying the usual quantum conditions instead of c-numbers, so that their Hamiltonian equations of motion become true quantum equations. The Hamiltonian function will now provide a Schrödinger wave equation, which must be solved and interpreted in the usual manner. The interpretation will give not merely the probable number of systems in any state, but the probability of any given distribution of the systems among the various states, this probability being, in fact, equal to the square of the modulus of the normalised solution of the wave equation that satisfies the appropriate initial conditions. We could, of course, calculate directly from elementary considerations the probability of any given distribution when the systems are independent, as we know the probability of each system being in any particular state. We shall find that the probability calculated directly in this way does not agree with that obtained from the wave equation except in the special case when there is only one system in the assembly. In the general case it will be shown that the wave equation leads to the correct value for the probability of any given distribution when the systems obey the Einstein-Bose statistics instead of being independent.
We assume the variables br , ih br* of § 2 to be canonical q-numbers satisfying the quantum conditions or
and
The transformation equations (8) must now be written in the quantum form
(10)
in order that the Nr, θr may also be canonical variables. These equations show that the Nr can have only integral characteristic values not less than zero,kb which provides us with a justification for the assumption that the variables are q-numbers in the way we have chosen. The numbers of systems in the different states are now ordinary quantum numbers.
The Hamiltonian (7) now becomes
(11)
in which the Hrs are still c-numbers. We may write this F in the form corresponding to (9)
(11ʹ)
in which it is again composed of a proper energy term Σr Wr Nr and an interaction energy term.
The wave equation written in terms of the variables Nr iskc
(12)
where F is an operator, each θr occurring in F being interpreted to mean ih ∂/∂Nʹr. If we apply the operator e±iθr/h to any function f(Nʹ1, Nʹ2, ... Nʹr, ...) of the variables Nʹ1 , Nʹ2 , ... the result is
If we use this rule in equation (12) and use the expression (11) for F we obtainkd
(13)
We see from the right-hand side of this equation that in the matrix representing F, the term in F involving ei(θr−θs)/h will contribute only to those matrix elements that refer to transitions in which Nr decreases by unity and Ns increases by unity, i.e., to matrix elements of the type F(Nʹ1, Nʹ2, ... Nʹr, ... Nʹs; Nʹ1, Nʹ2 ...Nʹr − 1 ... Nʹs + 1 ...). If we find a solution ψ (Nʹ1, Nʹ2 ...) of equation (13) that is normalised [i.e., one for which ΣNʹ1, Nʹ2 .... |ψ(Nʹ1, Nʹ2 ...)|2 = 1] and that satisfies the proper initial conditions, then |ψ (Nʹ1, Nʹ2 ...)|2 will be the probability of that distribution in which Nʹ1 systems are in state 1, Nʹ2 in state 2, ... at any time.
Consider first the case when there is only one system in the assembly. The probability of its being in the state q is determined by the eigenfunction ψ (Nʹ1, Nʹ2, ...) in which all the Nʹ’s are put equal to zero except Nʹq, which is put equal to unity. This eigenfunction we shall denote by ψ {q}. When it is substituted in the left-hand side of (13), all the terms in the summation on the right-hand side vanish except those for which r = q, and we are left with
which is the same equation as (5) with ψ {q} playing the part of bq . This establishes the fact that the present theory is equivalent to that of the preceding section when there is only one system in the assembly.
Now take the general case of an arbitrary number of systems in the assembly, and assume that they obey the Einstein-Bose statistical mechanics. This requires that, in the ordinary treatment of the problem, only those eigenfunctions that are symmetrical between all the systems must be taken into account, these eigenfunctions being by themselves sufficient to give a complete quantum solution of the problem.ke We shall now obtain the equation for the rate of change of one of these symmetrical eigenfunctions, and show that it is identical with equation (13).
If we label each system with a number n, then the Hamiltonian for the assembly will be HA = ΣnH (n), where H(n) is the H of § 2 (equal to H0 + V) expressed in terms of the variables of the nth system. A stationary state of the assembly is defined by the numbers r1, r2 ... rn ... which are the labels of the stationary states in which the separate systems lie. The Schrödinger equation for the assembly in a set of variables that specify the stationary states will be of the form (6) [with HA instead of H], and we can write it in the notation of equation (5) thus:—
(14)
where HA (r1r2 ...; s1s2...) is the general matrix element of HA [with the time factor removed]. This matrix element vanishes when more than one sn differs from the corresponding rn; equals Hrmsm sm when sm differs from rm and every other sn equals rn; and equals Σn Hrnrn when every sn equals rn. Substituting these values in (14), we obtain
(15)
We must now restrict b (r1r2 ...) to be a symmetrical function of the variables r1, r2 ... in order to obtain the Einstein-Bose statistics. This is permissible since if b (r1 r2 ...) is symmetrical at any time, then equation (15) shows that ḃ(r1r2 ...) is also symmetrical at that time, so that b (r1r2 ...) will remain symmetrical.
Let Nr denote the number of systems in the state r. Then a stationary state of the assembly describable by a symmetrical eigenfunction may be specified by the numbers N1, N2 ... Nr ... just as well as by the numbers r1, r2 ... rn..., and we shall be able to transform equation (15) to the variables N1, N2 .... We cannot actually take the new eigenfunction b (N1, N2 ...) equal to the previous one b (r1r2 ...), but must take one to be a numerical multiple of the other in order that each may be correctly normalised with respect to its respective variables. We must have, in fact,
and hence we must take |b(N1, N2 ...)|2 equal to the sum of |b(r1r2 ...)|2 for all values of the numbers r1, r2 ... such that there are N1 of them equal to 1, N2 equal to 2, etc. There are N ! / N1 ! N2 ! ... terms in this sum, where N = Σr Nr is the total number of systems, and they are all equal, since b(r1r2 ...) is a symmetrical function of its variables r1, r2 .... Hence we must have
If we make this substitution in equation (15), the left-hand side will become ih (N1 ! N2 ! ... /N !) b(N1, N2 ...). The term Hrmsm b(r1r2 ... rm−1smrm +1 ...) in the first summation on the righthand side will become
(16)
where we have written r for rm and s for sm. This term must be summed for all values of s except r, and must then be summed for r taking each of the values r1, r2 .... Thus each term (16) gets repeated by the summation process until it occurs a total of Nr times, so that it contributes
to the right-hand side of (15). Finally, the term Σn Hrnrn b (r1, r2 ...) becomes
Hence equation (15) becomes, with the removal of the factor (N1 ! N2 ! .../N !) ,